In the 20th minute, Green put Allsopp one-on-one, but Neil Moss did well to keep out his shot.
Bournemouth were making the City defence work but Boaz Myhill barely had a clear shot to save.
Green came close to getting another before half time but Moss tipped it over to keep the score at 1-0.
It looked like curtains for the visitors on the hour mark when they were reduced to ten men and facing another penalty. Moss had come out of his area only to be tackled by Nick Barmby. He crossed in for Allsopp, but he was tugged down by Neil Young in the six yard box with the open goal in front of him. Referee Mr. Prosser awarded City a second penalty and showed Young the red card, but Moss dived left to keep out Green's penalty.
Moss again denied City by getting his fingertips to Stuart Elliott's thunderous 20-yard strike, touching it onto the crossbar.
Barmby had a great chance to get his first league goal in black and amber, but couldn't get past Moss with a one-on-one.
City's inability to finish off the game with a second goal made for a tense finish, but in truth the Cherries were unable to test Boaz Myhill.
City line-up (4-4-2): Myhill; Hinds, Joseph (Cort 71), Delaney, Dawson; Green, Ashbee, Lewis, Elliott; Allsopp (Walters 65), Barmby (Wilbraham 82).
Un-used subs: Duke, France.